The distribution of \(p\) -homodyne outcomes consists of a periodic arrangement of Gaussian distributions all with covariance \(\widetilde{\mathbf {\Sigma }}_p\)  on \(N\)  modes, each Gaussian centred at a point \(\mathbf {n}\sqrt{\pi }\)  where \(\mathbf {n}\)  are integer valued vectors from a set that corresponds to the ideal state of the qubits.
Suppose we obtain the values \(\mathbf {p}\)  after the homodyne measurements.
If we assume \(\mathbf {p}\)  could have resulted from a Gaussian distribution centered at any of the lattice points \(\mathbf {n}\) , then the so-called responsibility [1]} of a given lattice point for the result \(\mathbf {p}\)  is given by:
\(r(\mathbf {n}) = \exp \left[-\frac{1}{2}(\mathbf {n}\sqrt{\pi }-\mathbf {p})^T \widetilde{\mathbf {\Sigma }}_p^{-1}(\mathbf {n}\sqrt{\pi }-\mathbf {p})\right].\) 
